Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











Cosh




Mathematica Notation

Traditional Notation









Elementary Functions > Cosh[z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Involving exponential and algebraic functions > Involving exp and algebraic functions > Involving (a z+b)beta dzcosh(c z+e)





http://functions.wolfram.com/01.20.21.0401.01









  


  








Input Form





Integrate[(a z + b)^\[Beta] E^(p z) Cosh[c z + e], z] == (-(1/(2 a))) E^(-e - (b (c + p))/a) (b + a z)^(1 + \[Beta]) (E^((2 b c)/a) ExpIntegralE[-\[Beta], ((c - p) (b + a z))/a] + E^(2 e) ExpIntegralE[-\[Beta], -(((c + p) (b + a z))/a)])








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["a", " ", "z"]], "+", "b"]], ")"]], "\[Beta]"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", "z"]]], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c", " ", "z"]], "+", "e"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List["2", " ", "a"]]]]], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "e"]], "-", FractionBox[RowBox[List["b", " ", RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]]]], "a"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]], RowBox[List["1", "+", "\[Beta]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["2", " ", "b", " ", "c"]], "a"]], " ", RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["-", "\[Beta]"]], ",", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["c", "-", "p"]], ")"]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]]]], "a"]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e"]]], " ", RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["-", "\[Beta]"]], ",", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]]]], "a"]]]]], "]"]]]]]], ")"]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> &#8747; </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> &#946; </mi> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> p </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> e </mi> <mo> + </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mi> e </mi> </mrow> <mo> - </mo> <mfrac> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> a </mi> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> &#946; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mi> a </mi> </mfrac> </msup> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> E </mi> <annotation encoding='Mathematica'> TagBox[&quot;E&quot;, ExpIntegralE] </annotation> </semantics> <mrow> <mo> - </mo> <mi> &#946; </mi> </mrow> </msub> <mo> ( </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> - </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> a </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> e </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> E </mi> <annotation encoding='Mathematica'> TagBox[&quot;E&quot;, ExpIntegralE] </annotation> </semantics> <mrow> <mo> - </mo> <mi> &#946; </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> + </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> a </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <ci> &#946; </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> p </ci> <ci> z </ci> </apply> </apply> <apply> <cosh /> <apply> <plus /> <ci> e </ci> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> e </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <apply> <plus /> <ci> c </ci> <ci> p </ci> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <ci> &#946; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <ci> c </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> ExpIntegralE </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#946; </ci> </apply> <apply> <times /> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> </apply> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> </apply> </apply> <apply> <ci> ExpIntegralE </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#946; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> c </ci> <ci> p </ci> </apply> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["a_", " ", "z_"]], "+", "b_"]], ")"]], "\[Beta]_"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["p_", " ", "z_"]]], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c_", " ", "z_"]], "+", "e_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "e"]], "-", FractionBox[RowBox[List["b", " ", RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]]]], "a"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]], RowBox[List["1", "+", "\[Beta]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["2", " ", "b", " ", "c"]], "a"]], " ", RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["-", "\[Beta]"]], ",", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["c", "-", "p"]], ")"]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]]]], "a"]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e"]]], " ", RowBox[List["ExpIntegralE", "[", RowBox[List[RowBox[List["-", "\[Beta]"]], ",", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["c", "+", "p"]], ")"]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]]]], "a"]]]]], "]"]]]]]], ")"]]]], RowBox[List["2", " ", "a"]]]]]]]]]


Date Added to functions.wolfram.com (modification date)





2002-12-18